It is well known that various birefringent crystals can be used to double the frequency of an incoming laser beam. For efficient frequency conversion the phasematching condition must be satisfied. Based on the method of achieving the phasematching condition, second harmonic interactions are classified as Type I and Type II. In Type I phasematching, the input fundamental beam is linearly polarized and arranged to excite only one polarization eigenstate in the crystal in order to maximize doubling efficiency. In contrast for maximum efficiency in Type II phasematching, the input fundamental beam is arranged to excite both crystal polarization eigenstates equally. This result can be achieved by properly orienting linear or elliptically polarized light, or by using circularly polarized light.
For typical Type I phasematching, the wave vectors of the fundamental and second harmonic beams are collinear. The fundamental beam excites one polarization eigenstate and the second harmonic beam is generated in the orthogonal polarization eigenstate. In typical Type II phasematching the wave vectors of all beams are collinear, however, the situation is more complex since there are now two excited fundamental polarization eigenstates (denoted with subscripts 1 and 2) and a generated second harmonic beam in one of these polarization eigenstates.
In order to have efficient second harmonic generation for either a Type I or Type II nonlinear interaction the phasematching condition must be satisfied. The phasematching condition for Type II phasematching is expressed mathematically as the following: EQU k.sub.1.sup.(.omega.) +k.sub.2.sup.(.omega.) =k.sub.1.sup.(2.omega.) or k.sub.2.sup.(2.omega.) ( 1)
where k.sub.1,2.sup.(.omega.) denotes the wave vectors of the orthogonally polarized input beams and k.sub.1,2.sup.(2.omega.) denotes the wave vector of the second harmonic beam. For notational simplicity we will drop the superscript (.omega.) when referring to the fundamental beam in the following discussion. The polarization state of the second harmonic beam (1 or 2) depends on the sense of the crystal's birefringence.
Typically all the wave vectors in the above expression are collinear, however, the direction of power flow associated with each beam (Poynting vector S) is in general not collinear with its associated wave vector, k. More particularly, if the direction of travel of the beam is not parallel to the crystal's optical axes, the direction of the Poynting vector, S, will be different from the wave vector k. The walk-off angle .rho., between the two vectors is defined as follows: EQU .rho.=arctangent .vertline.D.times.E/(D.multidot.E) (2)
where D is the displacement vector and E is the electric field vector of the beam. Because of the crystal birefringence and dispersion, the index of refraction will be different for each beam and thus each beam will propagate differently. Accordingly, the walk-off angle .rho. associated with each of the Poynting vectors will in general be of different magnitude and in a different direction.
The phenomenon is illustrated in FIG. 1 wherein block 10 represents a crystal oriented for Type II phasematching. The incoming beam 20 enters the crystal normal to its face. The wave vectors k.sub.1 and k.sub.2 associated with the orthogonal displacement field vectors D.sub.1 and D.sub.2 travel collinearly with the input fundamental beam. However, each of the two Poynting vectors S.sub.1 and S.sub.2 separate from the associated wave vectors at the angles .rho..sub.1 and .rho..sub.2 respectively. When the light energy at the fundamental wavelength leaves the crystal the two Poynting vectors will be parallel but separated by a distance d. It should be remembered that the Poynting vectors define the actual direction of travel of the power of the beam while the wave vectors represent the direction orthogonal to beam phase fronts.
FIG. 2 illustrates the problem as it generally occurs in a KTP crystal oriented for Type II phasematching of a 1064 nm fundamental beam. In this case, the incoming fundamental beam 22 is oriented in a manner such that the wave vector k.sub.2 (associated with the D.sub.2 polarization state) is aligned with the crystallographic axes such that there is no walk-off with the associated Poynting vector S.sub.2. However, due to the different index of refraction with respect to the D.sub.1 polarization state, there will be a non-zero walk-off angle .rho. between the k.sub.1 wave vector and the S.sub.1 Poynting vector. The generated second harmonic is in the subscript 1 polarization state and also experiences walk-off from its associated wave vector which is collinear with the two fundamental wave vectors. The prior literature often refers to the walk-off between second harmonic k.sub.1.sup.(2.omega.) and S.sub.1.sup.(2.omega.) Vectors as the walk-off angle associated with this nonlinear interaction. However, as discussed the situation is actually more complex since two of the three beams involved in the interaction experience some walk-off. When the energy flow leaves the KTP crystal, the two Poynting vectors (S.sub.1 and S.sub.2) will once again be parallel, but spaced apart a distance d. While the subject invention will be described with reference to the situation where only one of the two fundamental beam Poynting vectors experiences walk-off from the associated wave vector, it is equally applicable to the more general situation illustrated in FIG. 1.
The Poynting vector walk-off effect in KTP is well known and can be calculated. For example, in a KTP crystal where the beam is directed along the optimum phasematching angle of about 23.32 degrees for 1064 nm light, the walk-off angle .rho. between k.sub.1 and S.sub.1 will be about 0.2 degrees. Assuming a crystal length of 5 mm, and recalling that S.sub.2 experiences no walk-off, the separation d, or walk-off between the vectors upon exiting the crystal, will be about 17 microns. For the second harmonic beam, the walk-off angle between k.sub.1.sup.(2.omega.) and S.sub.1.sup.(2.omega.) is slightly different due to crystal dispersion, being 0.26.degree..
Poynting vector walk-off of all forms is undesirable in second harmonic interactions. Poynting vector walk-off of the orthogonally polarized fundamental beams is particularly undesirable in intracavity second harmonic generation because it leads to parasitic intracavity losses and reduces doubling efficiency. For example, the beam radius within a KTP crystal used in a typical diode-pumped intracavity frequency doubled system is about 50 microns. Assuming a Poynting vector walk-off of 17 microns, it can be seen that significant beam distortion and spatial variation of the polarization states can occur, leading to reduced efficiency.
Practitioners in the prior art addressed the problem in two ways. First, the length of the crystal was kept relatively short to minimize the separation d between the vectors upon exiting the crystal. Unfortunately, doubling efficiency is related to the length of the crystal so this approach will prevent higher efficiencies from being attained.
Another method of minimizing the effects of walk-off is to expand the diameter of the beam so that distortion effects are minimized. The latter approach also has drawbacks because for a given power, doubling efficiency is decreased when the spot size in the crystal is increased. Accordingly, it would be desirable to find an approach which minimizes the effects of Poynting vector walk-off without reducing doubling efficiency.
Another well known phenomenon is the refraction effect which occur when a beam enters an input face of a material at an oblique angle of incidence. More particularly, the wave vector k of a light beam will be refracted from the incoming path according to Snell's law which relates the angles of incidence and the angles of refraction with the index of refraction of the materials on either side of the interface.
As noted above, in the case of birefringent crystals, the index of refraction for the two polarization states will be different. Accordingly, if the input fundamental beam enters the crystal at an oblique angle of incidence, the wave vectors associated with the two polarization states (k.sub.1 and k.sub.2) will be refracted by different amounts. The magnitude of this effect is governed by the orientation of the interface relative to the crystals axes and the angle of incidence of the incoming light. In the prior literature, this effect is sometimes referred to as double refraction. Unfortunately, the term double refraction has also been used in the literature to refer to Poynting vector walk-off and the combination of these two distinct physical effects. Therefore, for clarity in this application, the effect will be referred to as wave vector double refraction.
FIG. 3 illustrates the effect of wave vector double refraction. In FIG. 3, the Poynting vectors are not illustrated. As can be seen, a fundamental beam of light 24 enters crystal 10 at an oblique angle of incidence .theta.. The two wave vectors k.sub.1 and k.sub.2 are refracted different amounts and this difference can be expressed as the intermediate angle .delta.. The magnitude of the difference in the amount of refraction associated with the two polarization states is given by the following formula: EQU .delta.=arcsin(sin .theta./n.sub.1)-arcsin(sin .theta./n.sub.2)(3)
where .theta. is the angle of incidence, n.sub.1 is the index of refraction for the D.sub.1 displacement field vector, n.sub.2 is index of refraction for the D.sub.2 displacement field vector and the index of refraction of the surrounding medium (air) is 1. As will be discussed below, by properly arranging the angle of incidence of the beam as well as the axes of the crystal with respect to the input face, wave vector double refraction can be used to compensate for Poynting vector walk-off.
In the prior art, much effort has been expended in determining the ideal phasematching angle of a crystal. More particularly, there is some loci of directions, which satisfy the phasematching condition. In general one of these directions has the largest effective nonlinear coefficient and is the direction along which an input beam can be passed in order to optimize the doubling efficiency of a crystal. As noted above, in KTP, which is the most common crystal used for Type II phasematching, the optimum phasematching angle is 23.32 degrees off the x-axis and 90 degrees off the z-axis for doubling 1064 nm light using collinear wave vectors.
In prior art doubling systems, the crystal has been fabricated such that a normal to the input face forms an angle 23.32 degrees off the x-axis and 90.degree. off the z-axis. The input beam is then directed into the crystal normal to the input face to insure that optimum phasematching is achieved. In this geometry there will be no wave vector double refraction. As will be seen below, the prior art geometry is changed in the subject invention, wherein the incoming beam is directed at an oblique angle of incidence to the input face to create wave vector double refraction effects which can then be used to compensate for the Poynting vector walk-off effects. In addition, the crystalline axes are oriented with respect to the input face to create the optimum phasematching angle with respect to the refracted beam travelling in the crystal.
For more complete background information on nonlinear optical interaction see Nonlinear Optics, P. G. Harper and B. S. Wherett, Eds. San Francisco, Calif.: Academic, 1977 pgs. 47-160.